540 Dr. C. Chree : Applications of 



If the pressure be II at 2=0, then at depth s- it is given by 



p-n+gftz) (17) 



For the change in the volume of the cylinder we have 

 by (2) 



MBv =z^\\gpzdxdydz — ff (Xx + py)pd$ — f f Ipdxdy, 



where I is the length of the cylinder, X, /a, the direction- 

 cosines of the outwardly drawn normal to the cylindrical 

 surface. The first double integral is taken over the cylin- 

 drical surface (or vertical faces, if the body is prismatic), the 

 second double integral over the lower face z = L 



If now tn be the perpendicular drawn from the axis of the 

 cylinder on the tangent at (x, y) to the horizontal section, we 

 have 



(Xx + fiy) dS = (vrds) dz, 



where ds is an element of the perimeter of the cross section. 

 Also 



\'uyds=2<Tj 



where a is the area of the cross section ; 



and over the lower face 



§lpdxdy = l{n+g$(l) \<r. 



Thus we obtain 



dUv= 9P (ll2)v-2v\n+g x (l)\-v\ll+g^{l-)}, 



or Bvlv={ffp(lj2)- 9 (<j>(l)+2 x (l))-m\/3k. . (18) 



This is true for all cases of continuous variation of density 

 with the depth. In the particular case dealt with in § 5, 

 where the change of density arises from the compressibility, 

 we have 



f(z)^p '{l + (U+g P< !z)/y\. 



Thus $ {£) ^Po'{2(i + n/F) + iffp^ia }, 



and after reduction we have 



M*= {fl>| -3II- m 'z(2 + 2j + |W!)}/3*. (19) 

 Treating as negligible powers of U/k' and gpol/tt above the 



